Edwin A. Abbott’s Flatland: A Romance of Many Dimensions is a classic in the mathematics community. On the surface, it is the account of a square confined to living in a two-dimensional world, Flatland. One evening the square is allowed to experience an all-too-brief tour of three-dimensional space. He then returns to Flatland to attempt to share “The Gospel of Three Dimensions.” The story has often served as an entertaining introduction to multi-dimensional geometry. Lindgren and Banchoff, the authors of this annotated edition, explain that Abbott would have been surprised that this use of his story is by far its primary legacy. For the book truly has many dimensions: “It is an extended metaphor expressed in the language of mathematics; it is a satirical commentary on Victorian society; it is a geometric version of Plato’s parable of the cave; it is an expression of religious principle; and it is an expression of Abbott’s theory that imagination is the basis of all knowledge.” The commentary and notes accompanying the text explain all of this and more.

The text of Flatland is placed on the left-hand pages of the book, with line numbers appearing in multiples of five in the left margin. The right hand pages then contain notes and commentary keyed to the line numbers of the text. The page arrangement and fonts are easy on the eyes. The only drawback is that a few of the lengthier notes are continued in an appendix.

The notes treat various topics. Some explain the mathematics appearing in the text. Some provide historical background, usually concerning either Victorian England or classical Greece. The authors view Abbott as having “devised an extended geometric metaphor by projecting late-Victorian England onto a two-dimensional space with a ‘civilization’ in various ways similar to that of classical Greece. Further, he heightened his satirical commentary on the present by making prominent in this imaginary civilization the very aspects of classical Greece that its Victorian apologists had rationalized away — for example, slavery, a rigid class system, misogyny, and ancient forms of social Darwinism.” These targets of Abbott’s satire receive considerable coverage in the commentary. The authors have been careful readers and students of the text for years. They even document a few mistakes on Abbott’s part. An example of such an oversight is a reference to the square’s “eyes,” when in fact he possesses only one eye.

Many of the authors’ notes concern linguistic features. Abbott wrote Flatland in an antiquated style, frequently mimicking, for example, Elizabethan English or the diction of the King James Bible. He relished wordplay, and often coined words for effect. Among his favorite devices were puns and double-entendres. With these, he often chose a word such that its context supports both its Victorian meaning and a more antiquated meaning. In other places, a word carries either a Victorian or a more antiquated meaning along with a mathematical meaning. It is all good fun, and the authors point out much that we readers would probably otherwise miss.

Indeed many of the notes are lexical in nature, providing information on words in the text. The most interesting of these notes delve into the etymologies of the words concerned. The majority, however, are simply translations or glosses of obsolete words or of words that are being used with an archaic meaning. Since Abbott wrote with an antiquated style, many of the authors’ glosses would have been helpful to the less learned in Abbott’s own time. It only adds to the charm of the book that even more of its vocabulary is now old-fashioned, as many additional words in the text carry a Victorian usage no longer current for us. The authors kindly gloss these words as well. It would seem, though, that for many readers some of the glosses are unnecessary. Three examples: “brethren: plural of brother,” “annals: historical records,” “save: with the exception of.” There are many like these. This “overglossing” makes the book exceptionally accessible, especially with an appendix that provides an index of all these translated words listed alphabetically, with the chapter and line number given for each word. This annotated version of Flatland is clearly designed to be enjoyed by a very wide audience.

Lindgren and Banchoff base their text on Abbott’s second edition. One criticism leveled against Flatland in a review following its initial publication claimed that the two-dimensional figures of Flatland would not be able to see each other, as they lacked height. Abbott countered this criticism in a preface to his second edition, along with a few additional paragraphs in the main text. Because Abbott’s preface is a retrospective on the events described in the text, Lindgren and Banchoff transfer it to the end of their edition, thus making it an epilogue. This seems appropriate, and it is the only significant change they make to Abbott’s work. The authors document exactly where in the text Abbott updated his story to respond to the criticism. They moreover include an appendix containing a copy of both the critical review and Abbott’s epistolary response to it.

Readers of this edition will become more familiar with Edwin A. Abbott the person. The multitude of notes explaining, among other things, Abbott’s wordplay, his grasp of classical Greek culture, and his critical views of Victorian society makes us almost feel like we are having a conversation with him. There is a recommended reading list for those who want to learn more about Abbott and the various aspects of Flatland. The list’s subheadings: “Abbott biography,” “Higher-dimensional geometry,” “Flatland general,” “Plato’s Cave,” “Classical Greece,” and “Victorian Britain.” An appendix includes a chronology of Abbott’s life, a “mathematical biography,” and a passage describing his efforts to further the education of girls.

For anyone who has not read Flatland, I highly recommend this edition. Abbott’s story can be enjoyed by those with a taste for any combination of mathematics, linguistics, social commentary, history, philosophy, religion, and faith. Lindgren and Banchoff’s notes and commentary deepen the enjoyment of these many dimensions of Abbott’s creation.

David A. Huckaby is an assistant professor of mathematics at Angelo State University.

Introduction William F. Lindgren and Thomas F. Banchoff; Part I. This World: 1. Of the nature of Flatland; 2. Of the climate and houses in Flatland; 3. Concerning the inhabitants of Flatland; 4. Concerning the women; 5. Of our methods of recognizing one another; 6. Of recognition by sight; 7. Concerning irregular figures; 8. Of the ancient practice of painting; 9. Of the universal colour bill; 10. Of the suppression of the chromatic sedition; 11. Concerning our priests; 12. Of the doctrine of our priests; Part II. Other Worlds: 13. How I had a vision of Lineland; 14. How in my vision I endeavoured to explain the nature of Flatland, but could not; 15. Concerning a stranger from Spaceland; 16. How the stranger vainly endeavoured to reveal to me in words the mysteries of Spaceland; 17. How the sphere, having in vain tried words, resorted to deeds; 18. How I came to Spaceland and what I saw there; 19. How, though the sphere showed me other mysteries of Spaceland, I still desired more; and what came of it; 20. How the sphere encouraged me in a vision; 21. How I tried to teach the theory of three dimensions to my grandson, and with what success; 22. How I then tried to diffuse the theory of three dimensions by other means, and of the result; Appendix A. Critical reaction to Flatland; Appendix B. Biography and chronology of Edwin A. Abbott; References and recommended reading.